Formal definition for sine and cosine

Question

I stumbled this proof for the period of $2\pi$ for sine and cosine on $\mathbb{R}$. The characterization of sine and cosine relies on these properties

$$\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$$ $$\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$$ $$\lim_{h \to 0} \frac{\sin(h)}{h} = 1$$

And the initial conditions $\cos(0) = 1$ and $\sin(0) = 0$. From this point, we can compute the derivatives for sine and cosine so that $(\cos(x))' = -\sin(x)$ and $(\sin(x))' = \cos(x)$ useful to derive the power series. It is no surprise that the power series yield two periodic functions with a period of $2\pi$ as they parametrize the unit circle.

Consider this circle with the bounded sector area in green:

Defining a totally new unit for angles (denoted $₳$) as a bounded sector area $A$ over the radius $r$ squared such as

$$₳ = \frac{A}{r^2}$$

We notice this new unit satisfies $360^\circ = 2\pi = \pi₳$ so we must naturally define two function in place of sine and cosine, namely $c$ and $s$ which are periodic of period $\pi$, but we see that this two functions also preserve the properties above

$$c(a + b) = c(a)c(b) - s(a)s(b)$$ $$s(a + b) = s(a)c(b) + c(a)s(b)$$ $$\lim_{h \to 0} \frac{s(h)}{h} = 1$$

And the initial conditions $c(0) = 1$ and $s(0) = 0$ are also satisfied. However, although our new unit for angle definition defines a complete revolution in a period of $\pi$, we would still get power series for a period of $2\pi$ which means two revolutions for one period.

Fundamentally, the natural unit would be twice the bounded sector area $A$ over the radius $r$ to have one revolution for one period.

So what really defines sine and cosine, the radian and the unit circle or the properties above?

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Please do not mark this as a duplicate, I have read other answers but none of them really solved my question.

Answer 1

Your $s$ function is basically $\sin 2x$, so that it has a period of $\pi$.

But $$\lim_{h\to 0}\frac{\sin2h}{h}=2\ne1,$$ so that property no longer holds.

As to "what really defines them", it doesn't matter. If you take the power series definition or the differential equation definition, it makes the analysis easy. If you appreciate the intuition more, start with the circle. When I teach my students, I start from the circle in my notes, and then in a remark I show that this is the same as the power series definition.

What's important isn't what defines them, but what characterises them, i.e., what properties uniquely capture their behaviour. You can start either with the properties you listed, or with the differential equation, or the power series definition, or $\Re e^{i\theta}$ and $\Im e^{i\theta}$ for instance. These all uniquely determine the standard $\sin$ and $\cos$ functions.

Answer 2

Combined as a complex relation, the first two equations read

$$e^{i(a+b)}=e^{ia}e^{ib},$$ which can be generalized as

$$e^{z_a+z_b}=e^{z_a}e^{z_b}$$ by introducing real parts.

Now, the functional equation

$$f(z_a+z_b)=f(z_a)f(z_b)$$ does define an exponential function. Anyway, this necessitates some smoothness condition on $f$ (which I don't know exactly).

The condition on the limit can be seen as a "normalization", which makes the base of the exponential the number $e$.